"Thus conscience does make cowards of us all" (3.1). Can you explain the context in which the passage occurs and what it reveals?

In the "to be, or not to be..." soliloquy, Hamlet is not feeling suicidal. He is merely pondering his own EXISTANCE. Killing one's self, and choosing to no longer exist are two entirely separate subjects. Hamlet, here, is merely second guessing himslef. He is not a man of action, and his over analysis of nearly every situation is his downfall. The "conscience" he speaks of here is, obviously, his own. His over thinking is making him cowardly in the act avenging his father, and murdering Claudius. While the opportunity has presented itself on a few occasions (one where Hamlet thinks Claudius is praying, and decides to wait to kill him while he is sinning), his conscience has overtaken his ability to act in a rash way. In his "turning pale", he displays his lack of courage, a characteristic which is father had plenty of.

"To be, or not to be..." refers to his thoughts of existance, where the "conscience does make cowards of us all..." part of the soliloquy moves away from pondering the afterlife, and more towards his own actions (or lack there of) and his own mental process. His conscience is preventing him from killing Claudius, not himself.